Explain how denominator zeros relate to vertical asymptotes.
If $x = a$ makes the denominator $q(x)$ zero and the numerator $p(x)$ non-zero, then there's a potential vertical asymptote at $x = a$.
What happens if a zero appears in both the numerator and denominator?
It might indicate a 'hole' in the graph rather than a vertical asymptote.
Explain the impact of multiplicity on vertical asymptotes.
If the multiplicity of a zero in the denominator is greater than in the numerator, there's a vertical asymptote. Higher difference means faster approach to infinity.
Describe the behavior of a function near a vertical asymptote.
As $x$ approaches a vertical asymptote from the left or right, the function approaches either positive or negative infinity.
How do limits relate to vertical asymptotes?
One-sided limits approaching a vertical asymptote will tend towards $\infty$ or $-\infty$.
Define a vertical asymptote.
A vertical line that the graph of a function approaches but never touches, where the function's value goes to infinity (or negative infinity).
What is a rational function?
A function of the form $r(x) = \frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions.
Define the multiplicity of a zero.
The number of times its corresponding factor appears in the polynomial.
What is a 'hole' in a rational function?
A point where both the numerator and denominator of a rational function are zero. It is a removable discontinuity.
How do you find vertical asymptotes of a rational function?
1. Factor the numerator and denominator. 2. Identify zeros of the denominator. 3. Check if those zeros are also zeros of the numerator. 4. If a zero is only in the denominator, there's a vertical asymptote there.
How do you determine the behavior of a function near a vertical asymptote?
1. Find the vertical asymptote. 2. Evaluate the one-sided limits as $x$ approaches the asymptote from the left and right. 3. Determine if the function approaches $\infty$ or $-\infty$ from each side.
How do you determine if a rational function has holes?
1. Factor the numerator and denominator. 2. If a factor cancels out from both, there is a hole at the x-value that makes that factor zero.
Given $r(x) = \frac{x^2 - 1}{x^2 - 2x + 1}$, find the vertical asymptote(s).
1. Factor: $r(x) = \frac{(x-1)(x+1)}{(x-1)(x-1)}$. 2. Simplify: $r(x) = \frac{x+1}{x-1}$. 3. Vertical asymptote: $x=1$.
